We know if two RVs are marginally Gaussian and uncorrelated, they can be not jointly Gaussian. How about two correlated RVs which are marginally Gaussian? Are they Jointly Gaussian or no, there are occasions when they are not Jointly Gaussian?
Suppose $U,V$ are i.i.d. $N(0,1).$ Let $(X,Y)=(U,U)$ if $U>0$ and let $(X,Y)=(-|U|,-|V|)$ if $U\le0.$
Or let $Z\sim N(0,1)$ and let $(X,Y)=(Z,Z)$ if $|Z|>1$ and let $(X,Y)=(Z,2-Z)$ if $|Z|\le 1.$
In both cases, $X$ and $Y$ are marginally gaussian, correlated, but not jointly gaussian.